Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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1answer
422 views

dirichlet problem for laplace's equation

How can we show that a Dirichlet problem for Laplace's equation in a finite region has a unique solution. Usually we can consider u2 - u1, a difference in values.
11
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0answers
787 views

Kato inequality [closed]

For any real-valued smooth function $u$, we have the Kato inequality $|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$, which holds when $|\nabla u|\neq0$. If moreover $u$ ...
1
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1answer
129 views

harmonic function

Let $f$ be a real function with $\Delta f=0$ on an open ball $B_{2n}(y)\subset\mathbb{R}^N$. How would I show $$\int\limits_{B_n(y)}|Df|^2(z)dz\leq Cn\int\limits_{\partial B_n(y)}|Df|^2(z)d\sigma(z)$...
3
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1answer
154 views

property of harmonic functions

If a real-valued function $u$ is harmonic on a ball $B_{2r}(x)$ in $\mathbb{R}^n$, how would one show that $$\sup_{B_r(x)}u^2\leq\frac{2^n}{|B_{2r}(x)|}\int_{B_{2r}(x)}u^2(y) dy$$
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0answers
3k views

Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the Cauchy–...
11
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2answers
5k views

Logarithm of absolute value of a holomorphic function harmonic?

Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$. Is it true that $z\mapsto \log(|f(z)|)$ is ...
2
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1answer
725 views

Mean Value Property of Harmonic Functions: $ \lim_{t \to 0^+ } \frac{1}{n\alpha(n) t^{n-1}} \int_{ \partial{B(x,t)}} u(y) dS \ = u(x) $

I'll only include the step that throws me off unless more info is requested, but this is from LC Evans PDEs book: $$ \displaystyle \lim_{t \to \, 0^+ } \left[ \frac{1}{n\,\alpha(n) \, t^{n-1}} \int_{ ...
3
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2answers
1k views

Mean-value formula for inhomogeneous harmonic functions

I am working on Evans' PDE textbook problems, but I am stuck with the following problem about modification of the proof of the mean-value formula for harmonic functions. I cannot really see how to ...
1
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0answers
163 views

$|Du|^2$ is subharmonic if $u$ is harmonic.

In Evan's textbook "Partial Differential Equation", question 5 in section 2.5 says "$|Du|^2$ is subharmonic if $u$ is harmonic.". This can be easily proven, but do we really need the derivative $D$? I ...
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2answers
2k views

I need to prove that this function is harmonic! [Solved]

I need to prove that $u:\mathbb{R}\times(-\frac{\pi}{2},\frac{\pi}{2})\rightarrow\mathbb{R}$ $$u(x,y)=\sum_{n \ \text{ is odd}}\cos(ny)e^{n(x-n)}$$ is harmonic. I have no idea which theorem or ...
5
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1answer
679 views

Laplace's Equation in Polar Coordinates

I am trying to express Laplace's equation in terms of polar coordinates. That is, $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\\ x=r\cos\theta,\\ y=r\sin\theta. $$ My book ...
2
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1answer
276 views

Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.

Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic. I need help to write a rigorous proof. Thank you
3
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1answer
2k views

Laplace's Equation with One Inhomogeneous Boundary Condition

While solving Laplace's equation, $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0, $$ with Dirichlet boundary conditions $$\begin{align} u(x,0)&=f_1(x),\\ u(x,b)&=0,\\...
3
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3answers
415 views

Stuck on Laplace's Equation

I am trying to solve the following: $-\theta_{yy}-\theta_{xx}=0$ $\theta(0,y)=-1$ $\theta(1,y)=1$ $\theta_y(x,0)=1$ $\theta_y(x,1)=0$ I can separate this into two different ...
6
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1answer
752 views

Interior gradient bound

I would like some help with the following problem (Gilbarg/Trudinger, Ex. 2.13): Let $u$ be harmonic in $\Omega \subset \mathbb R^n$. Use the argument leading to (2.31) to prove the interior ...
1
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2answers
408 views

is the converse true: in a simply connected domain every harmonic function has its conjugate

The question is. Is the converse true: In a simply connected domain every harmonic function has its conjugate? I am not able to get an example to disprove the statement.
1
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1answer
89 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in $\Omega-\overline{...
1
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2answers
2k views

Proving the maximum principle for harmonic functions

I am in the middle of the proof of the maximum principle for harmonic functions. Given a harmonic function $u$ on the complex plane and $M_0\in \mathbb{C}$. Take $r>0$ and suppose there is an open ...
1
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2answers
491 views

Solving Poisson Equation

I have the following equation: $$ \begin{cases} H_{xx} + H_{yy} = xy \\ H(x,0) = 0 \\ H(x,1) = x \\ H(0,y) = 0 \\ H(1,y) = 0 \end{cases} $$ We want to solve this, so from inspection of ...
2
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1answer
2k views

Proving the mean value property of harmonic functions using distributions?

A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
7
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1answer
973 views

Complex and real forms of the Poisson integral formula

In my complex analysis book there is the expression $$\frac{1 - |z|^2}{|1 - \bar z e^{it}|^2}$$ and it says that when $z = re^{it}$, we can write the above expression as $$P_r(t) = \frac{1 - r^2}{1 - ...
1
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0answers
202 views

Notation in Gilbarg/Trudinger? [Section 2.8]

Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
2
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0answers
835 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A $C^0(\...
1
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0answers
722 views

Poisson equation on half-space

Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$. Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$. Define the function $F : \Bbb R^n \to \Bbb R$ by $$ F(x_1,\dotsc,...
3
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2answers
289 views

Boundary values of harmonic $u$ are $ u(e^{it}) = 5- 4 \cos t $; find $u(1/2)$ and $v(1/2)$.

My problem is the following: Let $u$ be a continuous real-valued function in the closure of the unit disk $\mathbb{D}$ that is harmonic in $\mathbb{D}$. Assume that the boundary values of $u$ are ...
2
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1answer
317 views

Harmonic function with bounded preimage

I recently saw a question here about bounded/unbounded preimages of a set under a harmonic function. The question asked did not seem to make sense as it was talking about harmonic functions on $\...
1
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1answer
164 views

Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded

Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded. I am not getting what theorem or result to apply. Could anyone help me?
2
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1answer
1k views

Maximum principle for harmonic functions in unbounded domains

We demonstrated the weak maximum principle for harmonic functions in bounded domains, proving it first considering the case u subharmonic, then approximating in this way: choose $v(x)=x_1^2-M$ so ...
3
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1answer
387 views

Two question on harmonic function

In a question paper I got the following two questions. $u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex ...
4
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0answers
1k views

Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
7
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3answers
446 views

Bounded, non-constant harmonic functions: how far are they from existing?

Let $f$ be a function that maps $\mathbb{Z}^2$ to $\mathbb{R}$ and consider the operator $T$ which replaces the value of $f$ at $(i,j)$ by the average of the values of $f$ at its four neighbors: $$ Tf(...
4
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2answers
318 views

harmonic function question

Let $u$ and $v$ be real-valued harmonic functions on $U=\{z:|z|<1\}$. Let $A=\{z\in U:u(z)=v(z)\}$. Suppose $A$ contains a nonempty open set. Prove $A=U$. Here is what I have so far: Let $h=u-v$...
14
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4answers
8k views

Composition of a harmonic function with a holomorphic function is still harmonic

If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ is a conformal mapping of a domain $D_0$ onto $D$, is $f \circ g$ harmonic in $D_0$? I noticed this question while reading ...
4
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2answers
410 views

Limit involving the laplacian

I'm trying to prove that if $\Omega$ is an open subset of $\mathbb{R}^n$ and $u$ a $C^2$ function then $$\lim_{r\to 0}\frac{2n}{r^2}\left(u(x)-\frac{1}{|\partial B_r(x)|}\int_{\partial B_r(x)}u(y)d\...
18
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2answers
2k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
4
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2answers
2k views

Green's function

Why does the Green's function $G(r,r_0)$ of the Laplace's equation $\nabla^2 u=0$, the domain being the half plane, is equal $0$ on the boundary? How can I interpret the Laplace's equation physically? ...
1
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2answers
296 views

Laplace's equation

I am very confused. So I have Laplace's equation $\nabla^2\phi(x,y)=0$ and B.C.'s $\phi(x,0)=f(x); \,\,\,\,\,\, \phi(x,1)\equiv0$ where I have to solve it by Fourier transform. So I take the ...
1
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1answer
299 views

Is it trivial that I will always find a solution to Laplace's equation via finite-difference method

I've followed the method explained in Numerical Recipes in C, chapter 19, to solve a elliptic equation: http://www.capca.ucalgary.ca/top/teaching/phys499+535/PHYS535/nrf90/pde-c19-0.pdf I'm ...
1
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3answers
335 views

How to prove the asymptotic behavior of the fundamental solutions to the 2D and 3D Laplace's equations?

Consider the fundamental solution of Laplace's equation in 2d and 3d: $$\Phi(x,y):= \begin{cases} \frac{1}{2\pi}\ln\frac{1}{|x-y|},\quad x,y\in{\mathbb R}^2\\ \frac{1}{4\pi}\frac{1}{|x-y|},\quad x,y\...